Question

The amount of time that a 4th grader reads comic books in a week is normally distributed with a mean of 6 hours and a standard deviation of 2 hours. Suppose sixteen 4th graders are randomly chosen. What is the probability that the sample mean time for reading comic books per week is between 5 and 7 hours (round off to fourth decimal place)

Answer 1

Answer:

0.9544 = 95.44% probability that the sample mean time for reading comic books per week is between 5 and 7 hours.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

$\mu = 6, \sigma = 2, n = 16, s = \frac{2}{\sqrt{16}} = 0.5$

What is the probability that the sample mean time for reading comic books per week is between 5 and 7 hours

This is the pvalue of Z when X = 7 subtracted by the pvalue of Z when X = 5. So

X = 7

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{7 - 6}{0.5}$

$Z = 2$

$Z = 2$ has a pvalue of 0.9772

X = 5

$Z = \frac{X - \mu}{s}$

$Z = \frac{5 - 6}{0.5}$

$Z = -2$

$Z = -2$ has a pvalue of 0.0228

0.9772 - 0.0228 = 0.9544

0.9544 = 95.44% probability that the sample mean time for reading comic books per week is between 5 and 7 hours.